GCF & LCM
calculator
The greatest common factor simplifies fractions; the least common multiple lines up denominators and repeating schedules. Enter two or three numbers and get both, with the prime breakdown.
Two sides of divisibility
The GCF (greatest common factor) is the largest number dividing all your inputs — the tool that reduces 24/36 to 2/3. The LCM (least common multiple) is the smallest number they all divide into — the common denominator machine. They're linked by a tidy identity:
GCF = 2²×3 = 12 · LCM = 2³×3² = 72 · check: 12×72 = 24×36 ✓
How it computes
Rather than factoring (slow for big numbers), the GCF uses Euclid's algorithm — repeatedly replacing the larger number with the remainder of the division, one of the oldest algorithms still in daily use (~300 BC). The LCM then falls out of the identity above.
Real uses
GCF simplifies fractions and ratios; LCM finds when repeating cycles align — buses every 12 and every 18 minutes meet every 36. Adding fractions with unlike denominators is LCM's home turf.
GCF & LCM FAQ
Simplifying fractions and ratios: divide the numerator and denominator by their GCF to reach lowest terms. 24/36 ÷ 12 = 2/3.
Finding common denominators when adding fractions, and aligning repeating cycles — two events every 12 and 18 minutes coincide every 36 minutes (the LCM).
For two numbers, GCF × LCM = a × b. This calculator uses Euclid's algorithm for the GCF and derives the LCM from that identity.
Yes — add the optional third number. GCF and LCM both extend by applying the two-number operation repeatedly.